Viscous Equations Research Articles

The existence and asymptotic behavior of solutions to 3D viscous primitive equations with Caputo fractional time derivatives

On the one hand, the primitive three-dimensional viscous equations for large-scale ocean and atmosphere dynamics are commonly used in weather and climate predictions. On the other hand, ever since the middle of the last century, it has been widely recognized that the climate variability exhibits long-time memory. In this paper, we first prove the global existence of weak solutions to the primitive equations of large-scale ocean and atmosphere dynamics with Caputo fractional time derivatives. Then we establish the existence of an absorbing set, which is positively invariant. Finally, an attractor (strictly speaking, the minimal attracting set containing all the limiting dynamics) is constructed for the time fractional primitive equations, which means that the present state of a system may have long-time influences on the states in far future. However, there was no work on the long-time behavior of the time fractional primitive equations and we fill this gap in this paper.

Global solutions to the Cauchy problem of viscous shallow water equations in some classes of large data

Global solutions to the Cauchy problem of viscous shallow water equations in some classes of large data

Finite-dimensional global attractor for the three-dimensional viscous Camassa-Holm equations with fractional diffusion on bounded domains

Finite-dimensional global attractor for the three-dimensional viscous Camassa-Holm equations with fractional diffusion on bounded domains

Two approaches to instability analysis of the viscous Burgers' equation

Two approaches to instability analysis of the viscous Burgers' equation

Global ISS for the Viscous Burgers' Equation with Dirichlet Boundary Disturbances

Global ISS for the Viscous Burgers' Equation with Dirichlet Boundary Disturbances

Approximate Damped Oscillatory Traveling Wave in Viscous Compressible Fluid With Capillary Effect

We convert the viscous compressible fluid equations with capillary effect into the equivalent planar dynamical system and illustrate the global qualitative phase diagrams corresponding to its bounded orbits by the qualitative theory of ordinary differential equations. From the evolutionary relationship of bounded orbits in global qualitative phase diagrams and the correspondence between the bounded orbit and traveling wave, we discover the evolutionary rule of the profile of the traveling wave as the viscous coefficient varying and the forming reason for the damped oscillatory traveling wave. Further, we construct the analytical approximate explicit expression of the damped oscillatory traveling waves in accordance with the forming reason and prove its error decreases in exponential form. Finally, we compare the approximate damped oscillatory traveling wave with the numerical solution. The result shows they fit well.

The continuous dependence of the viscous Boussinesq equations uniformly with respect to the viscosity

The continuous dependence of the viscous Boussinesq equations uniformly with respect to the viscosity

A 3-D Viscous Vorticity Model for Predicting Turbulent Flows over Hydrofoils

This research addresses the demand for a computationally efficient numerical tool capable of predicting 3-D turbulent flows over 3-D hydrofoils, a critical step in ultimately addressing marine propeller or turbine performance. The related software development and its applications are conducted by employing the vorticity-based approach known as the viscous vorticity equation (VISVE). In particular, an existing 3-D laminar VISVE solver was modified in order to handle 3-D turbulent flow scenarios. The extension incorporates the k−ω SST model into the 3-D VISVE solver by using the finite volume method (FVM), thereby broadening its application to turbulent flows. The model was then tested in the case of turbulent flows over 3-D hydrofoils. The results were found to not be sensitive to either grid or time step size and to be in very good agreement with those obtained using a Reynolds-averaged Navier–Stokes (RANS) solver. This solver offers distinct advantages, including a significantly reduced computational domain size and reduced computational costs through its vorticity-based approach. Notably, turbulence concentration within boundary layers and free shear flows does not compromise the method’s computational efficiency. The simplified meshing process, which automatically generates the grids based on the number of panels on the hydrofoil, enhances accessibility for researchers and engineers.

A 17-point composite average-derivative optimal method for high-accuracy frequency-domain scalar wave modeling

The frequency-domain finite-difference (FD) technique, especially the average-derivative method (ADM), is widely utilized in wavefield modeling. However, conventional low-order FD techniques are insufficient for high-accuracy demands. Applying high-order FD operator and optimizing their coefficients are effective measures for improving the accuracy of seismic modeling. However, these approaches suffer from the limitations of constant density media and approximant perfectly matched layer (PML). To address this issue, we have developed a composite-ADM 17-point scheme for a scalar-wave equation. This novel scheme is based on the composition of ADM discretizations with different grid intervals, which successfully removes the above limitations and achieves high accuracy. To suppress the numerical dispersion, we take the L∞ norm to construct the objective function, which we minimize using the Grey Wolf Optimizer (GWO) and Particle Swarm Optimization (PSO) algorithms. Numerical analysis indicates that the composite-ADM greatly reduces phase velocity error compared with the traditional-ADM, requiring only 2.44 grid points per wavelength within a 1% phase velocity error. Using numerical examples in homogeneous and complex models, we demonstrate the accuracy and versatility of the composite-ADM 17-point scheme in variable density media with exact PML. Furthermore, we extend this optimal scheme to the case of 2D viscous and 3D scalar-wave equations, making it a more practical and versatile solution.

A Review on Burgers' Equations and It's Applications

This article presents a brief study on the review of the Burgers' equation. It also gives some concepts/ideas and techniques to solve Burgers' equation. Applying Burgers' equation to traffic flow requires concentrated effort for the solution. We develop our insights on how to obtain the Navier-Stokes equation through our inquiry into Burgers' equation. We also demonstrate how the Cole-Hopf transformation for the viscous Burgers' equation is derived. Finally, we use Burger's equation function as a model for the flow of traffic. Additionally, by employing the linear system method, we are able to obtain the answer to the one-way traffic flow problem. The Navier-Stokes equation has been derived to get in-viscid Burger's equation. The principle of Traffic flow and Navier-Stokes models have also been derived.

Boundedness of solutions for parabolic-elliptic predator-prey chemotaxis-fluid system with logistic source term

This paper considers the 2-species chemotaxis-Stokes system with competitive kinetics{(n1)t+u⋅∇n1=Δn1−χ∇⋅(n1∇w)+n1(λ1−μ1n1+an2),x∈Ω,t>0,(n2)t+u⋅∇n2=Δn2+ξ∇⋅(n2∇z)+n2(λ2−μ2n2−bn1),x∈Ω,t>0,wt+u⋅∇w=Δw−w+n2,x∈Ω,t>0,u⋅∇z=Δz−z+n1,x∈Ω,t>0,ut+∇P=Δu+(n1+n2)∇ϕ,x∈Ω,t>0,∇⋅u=0,x∈Ω,t>0 under no-flux boundary conditions for n1, n2, w and z in three-dimensional bounded domains and no-slip boundary conditions for u, this is∂n1∂ν=∂n2∂ν=∂w∂ν=∂z∂ν=0,u=0,x∈∂Ω,t>0, where χ>0, ξ>0, μ1≥0, μ2≥0, λ1≥0, λ2≥0, a≥0, b≥0 and ϕ∈W2,∞(Ω). This system is a coupled system of the chemotaxis equations and viscous incompressible fluid equations. Under appropriate assumptions, this problem exhibits a global classical bounded solution.

Global well-posedness and large-time behavior of solutions to the 3D inviscid magneto-micropolar equations with damping

We first establish the existence of global strong solutions to the 3D inviscid incompressible magneto-micropolar equations with a velocity damping in Sobolev spaces H K ( K ⩾ 3 ) , where the H 3 norm of initial data is small, but its higher order derivatives could be large. Combining the H ˙ − s norm ( 0 ≤ s < 3 2 ) or B ˙ 2 , ∞ − s norm ( 0 < s ≤ 3 2 ) of the initial magnetic field is finite with some tricky interpolation estimates, we show ‖ ∇ n B ( t ) ‖ L 2 ≤ C 0 ( 1 + t ) − n 2 − 3 2 p + 3 4 and two faster decay rates for the velocity and the micro-rotation field ‖ ∇ n ( U , w ) ( t ) ‖ L 2 ≤ C 0 ( 1 + t ) − n 2 − 3 2 p + 1 4 , which are shown to be the usual L p − L 2 ( 1 ≤ p ≤ 2 ) type of the optimal time decay rates for 3D inviscid incompressible magneto-micropolar equations with damping. We then conclude the damping term contributes to weaken the assumption of initial condition and enhance the decay rate of velocity compared to the classical incompressible viscous magneto-micropolar equations. Meanwhile, for the 3D inviscid compressible magneto-micropolar fluid, the damping term also has the same effect on the decay rate of the velocity.

Energy conservation of the resistive and the non‐resistive magnetohydrodynamic equations

In this paper, we provide sufficient conditions on the incompressible viscous resistive MHD equations and the viscous non‐resistive MHD equations to ensure the energy equality. Part of our results also holds for the ideal MHD equations.

A numerical method for analysis and simulation of diffusive viscous wave equations with variable coefficients on polygonal meshes

A numerical method for analysis and simulation of diffusive viscous wave equations with variable coefficients on polygonal meshes

Unconditionally energy-stable mapped Gegenbauer spectral Galerkin method for diffusive–viscous wave equation in unbounded domains

The diffusive–viscous wave equation plays a significant role in investigating the attenuation of seismic wave propagating in fluid-saturated medium. This paper focuses on the design of unconditionally energy-stable spectral method for the diffusive–viscous wave equation defined in unbounded domains Rd(d=2,3). We develop such a spectral method by using mapped Gegenbauer functions for the spatial approximation and Crank–Nicolson scheme for the temporal discretization. Then we show that the fully-discrete method satisfies the discrete energy-dissipation law without any restriction on the time step size. To achieve an efficient implementation, the matrix diagonalization procedure is employed to solve the linear systems for d=2,3. Finally, we carry out numerical examples in 3D case to demonstrate the good behavior of our method.

Dynamics through three-inlets of t-shaped ducts: Significance of inlet velocity on transient air and water experiencing cold fronts subject to turbulence

Efficiency, management, and control of air and water dynamics subject to cold fronts within ducts are among the problems that occur in the water industry and during the production of gases. This article presents the dynamics of unsteady air and water through a three-inlet t-shaped duct, emphasizing the problem of cold fronts. Attempts were made to remark on the nature of these fluids a few distances after the mixture (70 mm), before turning of the duct, and after turning the duct. The realizable k−ε viscous model and energy equation were solved numerically using Ansys Fluent 2022R1. Examination of the residual and mesh independency were considered for checking the convergence of results. Optimal eddies' average kinetic energy per unit mass was formed near the t-junction not because of the earlier formed cold fronts but due to the recirculation of pressure at the 90o bend that affect the velocity of cold fluid substance from the two adjacent inlets (i.e. inlet 2 and inlet 3). Considering the variation of inlet velocity, three cases of cold front formation was examined. When the entry velocities of cold air/water and warm air/water are the same, the formation of the cold front is invisible at the early stage despite the fact that air flows faster than water. A significant difference exists between the heat transfer rate at hot and cold inlets, but the mass transfer is uniform at the three inlets. The optimal temperature across the motion of both fluids occurs at the circular surface 30mm after the mixture. The duct's bent nature is a factor that greatly influences the mixture of cold and warm liquid substances.

A space-time pseudospectral method for solving multi-dimensional quasi-linear parabolic partial differential (Burgers') equations

The viscous Burgers' equation is a kind of quasi-linear important partial differential equation appeared in many fields. The nonlinear term contained in this equation results in difficulty in finding its high-precision numerical solution. In this paper, a space-time pseudospectral method for solving multi-dimensional Burgers' equations is proposed based on the Lagrange polynomials. The method is employed in time and space both at Chebyshev- Gauss- Lobbato(CGL) points. The spectral coefficients are found in such a way that the residual becomes minimum. The given problem is reduced to a system of nonlinear algebraic equations, which is solved by Newton-Raphson's method. Error bounds on discrete L2-norm and Sobolev norm (Hp) are presented. The computational experiments are carried out to corroborate the theoretical results and to compare the present method with existing methods in the literature. Moreover, some model examples of multi-dimensional Burgers' equations are tested and a detailed comparison of the proposed method with various other methods is also given. When using the proposed method, some interesting phenomenon appears with the coefficient of kinematic viscous.

A well-balanced and exactly divergence-free staggered semi-implicit hybrid finite volume / finite element scheme for the incompressible MHD equations

We present a new exactly divergence-free and well-balanced hybrid finite volume/finite element scheme for the numerical solution of the incompressible viscous and resistive magnetohydrodynamics (MHD) equations on staggered unstructured mixed-element meshes in two and three space dimensions. The equations are split into several subsystems, each of which is then discretized with a particular scheme that allows to preserve some fundamental structural features of the underlying governing PDE system also at the discrete level. The pressure is defined on the vertices of the primary mesh, while the velocity field and the normal components of the magnetic field are defined on an edge-based/face-based dual mesh in two and three space dimensions, respectively. This allows to account for the divergence-free conditions of the velocity field and of the magnetic field in a rather natural manner. The non-linear convective and the viscous terms in the momentum equation are solved at the aid of an explicit finite volume scheme, while the magnetic field is evolved in an exactly divergence-free manner via an explicit finite volume method based on a discrete form of the Stokes law in the edges/faces of each primary element. The latter method is stabilized by the proper choice of the numerical resistivity in the computation of the electric field in the vertices/edges of the 2D/3D elements. To achieve higher order of accuracy, a piecewise linear polynomial is reconstructed for the magnetic field, which is guaranteed to be exactly divergence-free via a constrained L2 projection. Finally, the pressure subsystem is solved implicitly at the aid of a classical continuous finite element method in the vertices of the primary mesh and making use of the staggered arrangement of the velocity, which is typical for incompressible Navier-Stokes solvers. In order to maintain non-trivial stationary equilibrium solutions of the governing PDE system exactly, which are assumed to be known a priori, each step of the new algorithm takes the known equilibrium solution explicitly into account so that the method becomes exactly well-balanced. We show numerous test cases in two and three space dimensions in order to validate our new method carefully against known exact and numerical reference solutions. In particular, this paper includes a very thorough study of the lid-driven MHD cavity problem in the presence of different magnetic fields and the obtained numerical solutions are provided as free supplementary electronic material to allow other research groups to reproduce our results and to compare with our data. We finally present long-time simulations of Soloviev equilibrium solutions in several simplified 3D tokamak configurations, showing that the new well-balanced scheme introduced in this paper is able to maintain stationary equilibria exactly over very long integration times even on very coarse unstructured meshes that, in general, do not need to be aligned with the magnetic field.

Ill-posedness issue for the 2D viscous shallow water equations in some critical Besov spaces

We study the Cauchy problem of the 2D viscous shallow water equations in some critical Besov spaces B˙p,12p(R2)×B˙p,q2p−1(R2). As is known, this system is locally well-posed for large initial data as well as globally well-posed for small initial data in B˙p,12p(R2)×B˙p,12p−1(R2) for p<4 and ill-posed in B˙p,12p(R2)×B˙p,12p−1(R2) for p>4. In this paper, we prove that this system is ill-posed for the critical case p=4 in the sense of “norm inflation”. Furthermore, we also show that the system is ill-posed in B˙4,112(R2)×B˙4,q−12(R2) for any q≠2.

Vanishing dissipation limit of solutions to initial boundary value problem for three dimensional incompressible magneto-hydrodynamic equations with transverse magnetic field

In this paper, we study the vanishing dissipation limit of solutions to the initial boundary value problem of three-dimensional incompressible viscous magnetohydrodynamic equations in the upper half space where the magnetic field is assumed to be transverse to the boundary. When the tangential component of the magnetic field is imposed the zero Neumann boundary condition, we can establish the uniform energy estimates of the solutions to the initial boundary value problem in the conormal Sobolev spaces even if the velocity field satisfies the no-slip boundary condition. Then the vanishing dissipation limit of solutions to the initial boundary value problem can be achieved in L∞ sense based on some compactness arguments. Compared with the previous results, our results show that the strong boundary layer can still be prevented by the transverse magnetic field even with the magnetic diffusion.